Problem: The average of the numbers 23 and $x$ is 27. What is the positive difference between 23 and $x$? (Recall that the average of two numbers is their sum divided by 2).
Explanation: To calculate the average of two numbers, we add them and divide the sum by $2$.  From the given information, we have the equation  \[\frac{23+x}{2}=27.\]  Multiplying both sides by $2$ yields \[23+x=54.\]  Thus, $x=54-23=31$.  Finally, the positive difference between $31$ and $23$ is \[31-23=\boxed{8}.\]